The free motion of a rocket-propelled body

Imagine a rocket-propelled body moving in a region where aerodynamic drag and lift and gravitational force may be neglected, i. e. in space remote from any planets, etc. At time t let the mass of the body plus unburnt fuel be M, and the speed of the body relative to some axes be V. Let the fuel be consumed at a rate of m, the resultant gas being ejected at a speed of v relative to the body. Further, let the total rearwards momentum of the rocket exhaust, produced from the instant of firing to time t, be /relative to the axes. Then, at time t, the total forward momentum is

H{ = MV-I (9.64)

At time (/ + 6t) the mass of the body plus unbumt fuel is (M – m6t) and its speed is (V + 6V), while a mass of fuel m6t has been ejected rearwards with a mean speed, relative to the axes, of (v — V — j6V). The total forward momentum is then

tf2 = {M – m8t){V + 8V) – rh6t{v – V-^6V)-I

Now, by the conservation of momentum of a closed system:

#, =#2

i. e.

MV – I =MV + M6V – mV6t – mStSV – mv6t + mV6t + ^mStSV-I

which reduces to

M6V — – mStSV — mvSt = 0 2

Dividing by 6t and taking the limit as 6t —> 0, this becomes

Note that this equation can be derived directly from Newton’s second law, force = mass x acceleration, but it is not always immediately clear how to apply this law to bodies of variable mass. The fundamental appeal to momentum made above removes any doubts as to the legitimacy of such an application. Equation (9.65) may now be rearranged as

dV _ m

Table 9.1

t (min)

M (1000 kg)

Acceleration (m s 2)

f (ms!)

x (km)

0

11

4.55

0

0

0.5

10

5.00

143

2.18

1

9

5.55

300

9.05

1.5

8

6.25

478

19.8

2

7

7.15

679

37.6

2.5

6

8.33

919

61.4

3

5

10.0

1180

92.0

3.5

4

12.5

1520

133

4

3

16.7

1950

185

4.5

2

25.0

2560

256

5

1

50.0

3600

342

5.5

1

0

3600

450

Substituting the above values into the appropriate equations leads to the final results given in Table 9.1. The reader should plot the curves defined by the values in Table 9.1. It should be noted that, in the 5 minutes of burning time, the missile travels only 342 kilometres but, at the end of this time, it is travelling at 3600 m s_ 1 or 13 000 km h-1. Another point to be noted is the rapid increase in acceleration towards the end of the burning time, consequent on the rapid percentage decrease of total mass. In Table 9.1, the results are given also for the first half-minute after all-burnt.